proof that the set of sum-product numbers in base 10 is finite

First, Wilson proved that 10m-1≤n (where m is the number of digits of n) and that

∑i=1mdi≤9⁢m

and

∏i=1mdi≤9m

. The only way to fulfill the inequality 10m-1≤9m⁢9⁢m is for m≤84.

Thus, a base 10 sum-product number can’t have more than 84 digits. From the first 1084integers, we can discard all those integers with 0’s in their decimal representation. We can further eliminate those integers whose product of digits is not of the form 2i⁢3j⁢7k or 3i⁢5j⁢7k.

Having thus reduced the number of integers to consider, a brute force search by computer yields the finite set of sum-product numbers in base 10: 0, 1, 135 and 144.